
TL;DR
This paper introduces flip signature as a new invariant for $D_{ fty}$-topological Markov chains, demonstrating its effectiveness in distinguishing non-conjugate systems with natural $D_{ fty}$-actions.
Contribution
The paper defines flip signature, proves it is a $D_{ fty}$-conjugacy invariant, and applies it to differentiate specific $D_{ fty}$-actions on symbolic systems.
Findings
Flip signature is a $D_{ fty}$-conjugacy invariant.
Ashley’s eight-by-eight and full two-shift are not $D_{ fty}$-conjugate.
Introduces $D_{ fty}$-shift equivalence and discusses Lind zeta function.
Abstract
A -topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group . It is defined by two zero-one square matrices and satisfying and . Flip signature is obtained from symmetric bilinear forms with respect to on the eventual kernel of . We modify Williams' decomposition theorem to prove flip signature is a -conjugacy invariant. We introduce natural -actions on Ashley's eight-by-eight and the full two-shift. The Flip signatures show that Ashley's eight-by-eight and the full two-shift equipped with the natural -actions are not -conjugate. We also discuss the notion of -shift equivalence and the Lind zeta function.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Markov Chains and Monte Carlo Methods · Cellular Automata and Applications
